The EU Agricultural Committee has called for "an amnesty of three years on cross-compliance penalties relating to electronic identification of sheep and goats" and Mr Moore has tabled a similar Parliamentary motion.
The committee argue that farmers need more time to become accustomed to the complex new technology and urged the EU Commission to conduct a thorough review of the EID legislation.
Liberal Democrat MSP Liam McArthur also tabled a similar motion at the Scottish Parliament in Edinburgh last week.
Mr Moore has worked closely with representatives of the Scottish farming industry on this issue in the past, and last year pressed Defra Ministers to support NFUS calls for a compromise solution on sheep tagging from the floor of the House of Commons. He has also raised the concerns of Borders hill farmers directly with EU agricultural officials during meetings in Brussels.
Mr Moore said: "Farmers groups from all over Europe warned that the complexity of the new EID rules would make it difficult for producers to avoid stiff cross-compliance penalties before they were introduced.
"A three year amnesty would offer Borders hill farmers time to make the significant changes in their businesses that the new legislation demands without facing the prospect of substantial fines.
"After another year in which many farmers have found it difficult to make ends meet, there is a clear need for officials to listen to producers and exercise a bit of common sense.
"It is essential that there is flexibility in the application of the new law until a full review of the changes is complete."
Roxburgh and Berwickshire MSP John Lamont is also keen to see a stay of execution before non-compliance hits farmers' pockets.
He said: "The introduction of sheep tagging has always been an unwelcome development for the Scottish sheep industry, and my colleagues and I will continue to press for more concessions to be granted to the industry, following the unnecessary introduction of this legislation.
"The new measures will be costly and very disruptive for farmers in the Borders to implement and it is important that they are allowed time to adjust to the new measures. That is why I think it would be sensible to allow an amnesty which would let Borders sheep farmers get on with adapting their business to the new measures without having to face the prospect of crippling fines.
"The news that the Scottish Government had also responded to pressure from the Conservatives and the NFUS to introduce critical control points which allow tags to be scanned at markets or abattoirs was also a small relief.
"If these unwelcome measures are to be implemented, it is important that disruption to the industry is kept to a minimum."